Flight envelope using LFR models design by flight point’s interpolation

Linear Quadratic Regulation (LQR)

The advantage of the Linear Quadratic Regulation (LQR) method is that it provides the smallest possible error to both inputs and outputs while minimizing the control effort; the error corresponds to the difference between the desired and obtained value for system input and output. In the case when the full states are measurable, the LQR method ensures a stable controller output for the nominal model, and it provides cross-terms in the flight dynamics equations. Consequently, it leads to a robust controller in the sense that the gain margin is infinite and the phase margin is greater than 60 degrees. It was illustrated in the literature by Boughari et al (2012) that LQR method has been used for the Stability Augmentation System (SAS) control, and applied on Hawker 800XP business aircraft. In addition, the Linear Quadratic Gaussian (LQG) method has been used in (Botez et al., 2001) bomber B-52 aircraft to alleviate the gust effects. The LQR method has also been used in a longitudinal attitude controller designed for B747 aircraft (Guilong et al., 2013), and in adaptive LQR gain scheduling control is designed for remotely controlled aircraft (Mukherjee et Pieper, 2000).

In order to obtain corresponding optimal state feedback gain K in the LQR control methodology, the objective function which represents the quadratic performance index function J must be defined. This means the appropriate Q and R weighting matrices need to be estimated by a trial and error method, or by relying on the designer’s knowledge until the desired response is found. In order to overcome the time-consuming LQR procedure, many algorithms were developed in the last decade to optimize the LQR weighting matrices searches. Use of stochastic searching as an optimization algorithm is one of the most popular methods that have been used recently; in (Wongsathan et Sirima, 2008) , (Wongsathan et Sirima, 2009), used stochastic search method to determine LQR weighting matrices to control an inverted pendulum, and then a triple inverted pendulum. Satisfactory results were obtained by comparison of the optimal Q and R matrices with the weighting matrices obtained through “trial and error”.

The optimized

LQR methodology using the Genetic Algorithm (GA) was applied on the buck converter to improve its voltage control response, and the distillation column control, respectively shown in (Poodeh et al., 2007) and (Jones et Hengue, 2009). In both of those cases, the control performances that is given by weighting matrices found with a GA search provided better results than those found experimentally. (Ghoreishi et Nekoui, 2012) used both the Genetic Algorithm (GA) and Particle Swarm Optimization (PSO) algorithms with the LQR optimization. Guo et al used optimal LQR weighting matrices analyses based on the Genetic Algorithm (GA) search (Guo et al., 2010). Stochastic search for optimal LQR control combined with integral quadratic constraints was investigated by (Lim et Zhou, 1999); while Xiong et al. (Xiong et Wan, 2010) used LQR method based on PSO algorithm for double inverted pendulum control. In (Yoon Joon et Kyung Ho, 1997), the authors investigated stochastic searching methods for the determination of the LQR weighting parameters used for nuclear reactor power control. In (Zhu et Li, 2003), the authors have used an iterative method for solving stochastic Riccati differential equations of the LQR problem. Unfortunately, the LQR control can only provide a stability augmentation to the system, in order to perform the tracking error; a classical control method is added by using a PID control.

The PID control gain can be tuned using an ad hoc method or can be optimized using a stochastic algorithm. As illustrated by (Mitsukura, Yamamoto et Kaneda, 1997), the tuning of PID gain parameters was based on the GA, and on Fuzzy Logic in (Hyung-Soo et al., 1999). Han, Luo et Yang (Han, Luo et Yang, 2005), used a nonlinear PID controller based on genetic tuning, while a self-tuning algorithm was investigated by the authors for linear PID controllers; this algorithm was based on frequency characteristics in (Chen, Wang et Wu, 2010). In (Chang-Hoon, Myung-Hyun et Ik-Soo, 1997), the authors have used the model identification by use of two Nyquist points to automate the PID controller tuning; in (Bandyopadhyay et Patranabis, 2001), an auto-tuning algorithm for PID controllers based on dead-beat format requirements was performed using the fuzzy inference method. From previous researches, we can deduce that these optimized LQR and PID algorithms were mainly used in chemical industries. There is a huge amount of flight tests to be managed in the aircraft control design, thus in the aerospace industry; for this reason there is a great need in the use of the optimization algorithms that can be performed on control parameters to meet the design requirements, and to save time, and thus money, which is a part of investigation in this thesis.

H-infinity Controller The aircraft’s safety is dependent on its controller, as the clearance authorities need to ensure that the controller operates properly through the specified flight envelope even in presence of uncertainties related to mass, center of gravity positions, and inertia variations. The control clearance process is a fastidious and expensive task, especially for modern aircrafts that need to achieve high performance (C. Fielding, 2002) . This process aims to prove that the stability, robustness and handling requirements are satisfied against any possible uncertainties. During the industrial clearance process, the selection of the appropriate control laws with sufficient robustness involves: the investigation of the closed-loop eigenvalues, the stability margins and the performance indices, in the presence of uncertainties. The resulting control laws are used further for the design of the Flight Control System (FCS). The aircraft controller determination is very complex. Nonlinear methods such as Fuzzy Logic and Neural Network methods have been applied for Aircraft Identification and Control (G. Kouba, 2009),(N. Boëly, 2009). The Non- Linear Hybrid Fuzzy Logic Control on a morphing wing was explored (Grigorie et al., 2012a),(Popov et al., 2010). Due to its complexity in the Aerospace Industry, the determination of the robust Flight Control System FCS is usually carried out using linear methods applied on linear models, and it is further validated on non-linear models. In the literature, many linear control methods were used to obtain a FCS by the combination of modern control LQR method, the classical PID control method, and evolutionary algorithms that were applied successfully on the whole flight envelope of the Cessna Citation X (Boughari. et al., 20014a). However, the use of the LQR method allowed the system stabilization, while the classical PID control method was used for the tracking problem. A FCS that stabilizes and can track the reference input while taking disturbances into account was obtained by using, the H-infinity linear method proposed by Zames in 1983 (Zames), that had gained popularity in guarantying system robustness in the presence of uncertainties.

The H-infinity method has been used in the Aeronautical industry to develop controllers with the aim to meet the required system specifications and needs. One of the most important aspects of this controller is the determination of the weighting functions (W? and W?), which are very important in the gains calculation. There is no specific methodology to determine these weighting functions. The literature points out that the weighting functions are determined using a trial-an-error methodology, or pure experience-based methods. Several applications of this control method have been incorporated in the aeronautical domain, mostly for fighter jets, where a scheduled H-infinity controller was used for VSTOL longitudinal control (Hyde et Glover, 1993), and it has as well been used for the lateral control of an F-14 (G.J.Balas, 1998). An H-infinity controller design with gain scheduling approach was successfully used on a flexible aircraft where the weighting functions were not optimized, but were determined using Engineering intuition (Aouf, Boulet et Botez, 2002). To overcome this lack of reference formulas, some guidelines were given in (Ciann-Dong, Hann-Shing et Shin-Whar, 1994b; Hu, Bohn et Wu, 1999) to determine these weighting functions. However, due to their trial and error nature the guidelines procedures may need many iterations to find acceptable results. Besides, the guidelines do not guarantee the fulfillment of the required control conditions. For this reason, a methodology to tune the weighting functions to meet the mandatory requirements is necessary. There exist several weighting optimization methods based on mathematical algorithms, in which trade-offs were established between maximizing the stability margin and minimizing the H-infinity norm of the closed loop transfer function (Lanzon, 2005). These algorithms often performed on frequency-dependent optimizations, in which the iteration process demanded a considerable amount of memory allocation.

To overcome this frequencydependent optimization memory, a state space weight optimization was developed in (Osinuga, Patra et Lanzon, 2012b). However, that algorithm does not guarantee a global minimum convergence, which could lead to a poor stability margin, that could have a negative effect on a system operating in a large envelope, such as an aircraft. A new and innovative methodology by taking advantage of both GA and DE algorithms to optimize the H-infinity weighting functions to develop a controller that satisfies the imposed dynamic specifications and the industrial needs is proposed in this thesis. This new approach can solve the clearance problem by reducing the complexity of needed calculations and their validation. However, this research confirms that optimization using the DE algorithm is more efficient and accurate than the optimization using the GA; Storn and Price (Storn et Price, 1997) have also shown the efficiency of the DE algorithm by the comparison of its results with genetic algorithm results. Many global optimizations based on evolutionary principles have been used in the Control Engineering field. In the Aeronautical field, aircraft trajectory optimizations based stochastic search, such as the Genetic Algorithm (GA) were performed on several civil aircrafts (Murrieta-Mendoza et Botez, 2015a),(Patrón et Botez, 2015) as well as parameters estimation methodologies were performed on autonomous air vehicles and in the flight testing of the aircraft intelligent flight controls (Mario, 1999),(Osinuga, Patra et Lanzon, 2012a). These new methodologies for the control of different parameters are applied in this thesis for the flight dynamics and control of the business aircraft Cessna Citation X model. All these methods were developed in this thesis with the aim of reducing the computational complexity, and thus their time of convergence while achieving very good results. The GA and the Differential Evolution (DE) algorithms were selected to optimize the weighting function parameters.

Table des matières

INTRODUCTION
0.1 Problem Statement
0.2 Objectives
0.3 Methodology
0.3.1 Cessna Citation X business aircraft model
0.3.2 Aircraft dynamics
0.3.3 Flight envelope using LFR models design by flight point’s interpolation
0.3.4 Stability analysis interface
CHAPITRE 1 LITTERATURE REVIEW
1.1 Aircraft Flight Control System
1.2 Flight Control Optimization
1.3 Linear Quadratic Regulation (LQR)
1.4 H-infinity Controller
1.5 Aircraft Clearance Criteria
1.6 Linear Stability Criteria
1.7 Cessna Citation X Clearance Criteria Evaluation
CHAPITRE 2 APPROACH AND ORGANIZATION OF THE THESIS
CHAPITRE 3 CESSNA CITATION X BUSINESS AIRCRAFT AEROELASTIC STABILITY FLIGHT ENVELOPE ANALYSIS USING LFR MODELS – USING A NEW GUI FOR THE EASY MANIPULATION OF LFRs
3.1 Introduction
3.2 Cessna Citation X Business Aircraft Modeling
3.3 Linear Fractional Representation (LFR)
3.3.1 LFR modeling using Trends and Bands method
3.3.2 Normalization
3.3.3 The Graphical User Interface GUI
3.4 Stability Analysis
3.4.1 Lyapunov stability
3.4.2 Quadratic Stability
3.4.3 Resolution Method
3.4.3.1 Wang-Balakrishnan method
3.4.4 Stability analysis interface
3.5 Analysis of Results
3.5.1 LFR results validation
3.5.2 Stability analysis results
3.5.2.1 Results of the aircraft longitudinal model stability analysis
3.6 Conclusion
CHAPITRE 4 NEW METHODOLOGY FOR OPTIMAL FLIGHT CONTROL USING DIFFERENTIAL EVOLUTION- APPLICATION TO THE CESSNA CITATION X AIRCRAFT – VALIDATION ON AIRCRAFT RESEARCH FLIGHT LEVEL D SIMULATOR
4.1 Introduction
4.2 Problem Statement
4.2.1 Aircraft control architecture using LQR and PI
4.2.2 Cessna Citation X business aircraft
4.2.3 Aircraft, actuators and sensors dynamics
4.2.3.1 Aircraft dynamics
4.2.3.2 Actuators and sensors dynamics
4.3 Flight Conditions Interpolation
4.4 Design Specifications and Requirements
4.5 Differential Evolution
4.5.1 Initialisation phase
4.5.2 Mutation
4.5.3 Crossover
4.5.4 Selection
4.5.5 Iteration
4.6 Linear Quadratic Regulation (LQR) Method
4.7 Tracking Control with PI Optimization
4.8 DE Algorithm for Solving the LQR-PI Problem
4.8.1 Objective function
4.9 Simulation Results Analysis
4.9.1 Results validation
4.9.1.1 Linear validation
4.9.1.2 Nonlinear validation
4.10 Conclusion
CHAPITRE 5 FLIGHT CONTROL CLEARANCE OF CESSNA CITATION X USING EVOLUTIONARY ALGORITHMS
5.1 Introduction
5.2 Cessna Citation X Business Aircraft
5.2.1 Aircraft dynamics
5.2.2 LFR models design by flight point’s interpolation
5.2.3 Flying quality’s level 1
5.3 H-infinity Theory
5.3.1 Definition of the standard H-infinity robust control problem
5.3.2 Definition of the mixed sensitivity H-infinity problem
5.4 Differential Evolution and Genetic Algorithms
5.4.1 Objective Function for DE algorithm and GA
5.4.2 Differential Evolution algorithm
5.4.2.1 Initialization phase
5.4.2.2 Mutation
5.4.2.3 Crossover
5.4.2.4 Selection
5.4.2.5 Iteration
5.4.3 Genetic Algorithm applied to the H-infinity method
5.5 Presentation of Results
5.5.1 GA and DE algorithm optimization results
5.5.2 Results for 72 flight conditions
5.5.3 Non-linear validation
5.5.4 Robustness analysis of H-infinity controller
5.6 Conclusion
CHAPITRE 6 OPTIMAL CONTROL NEW METHODOLOGIES VALIDATION ON THE RESEARCH AIRCRAFT FLIGHT SIMULATOR OF THE CESSNA CITATION X BUSINESS AIRCRAFT
6.1 Introduction
6.2 Cessna Citation X Aircraft, Actuators and Sensors Dynamic
6.2.1 Aircraft dynamics
6.2.2 Actuators and sensors dynamics
6.3 Flight Controller
6.4 Clearance Criteria
6.4.1 Linear stability and Eigenvalue analysis
6.4.2 Linear, Nonlinear handling qualities, and Nonlinear analysis
6.4.3 Pitch control , and Rapid roll
6.5 Analysis of Results
6.5.1 Stability analysis results
6.5.2 Eigenvalue results
6.5.3 Handling qualities analysis results
6.5.4 Nonlinear analysis results
6.6 Conclusion
DISCUSSION OF RESULTS
CONCLUSION AND RECOMENDATIONS
APPENDIX A
LIST OF REFERENCES

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